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arXiv · 2212.08428

On reduction numbers and Castelnuovo-Mumford regularity of blowup rings and modules

Cleto B. Miranda-Neto·Douglas S. Queiroz
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Abstract

We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing for the setting relative to a Cohen-Macaulay module). We derive a number of applications, including a characterization of (polynomial) ideals of linear type, progress on the theory of generalized Ulrich ideals, and improvements of results by other authors.

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Cleto B. Miranda-Neto, Douglas S. Queiroz. 2022-12-16. On reduction numbers and Castelnuovo-Mumford regularity of blowup rings and modules. https://arxiv.org/abs/2212.08428

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